Theorem ssun1 3106

Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun1	|- A C_ ( A u. B )

Proof of Theorem ssun1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 633	. . 3 |- ( x e. A -> ( x e. A \/ x e. B ) )
2		elun 3084	. . 3 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
3	1, 2	sylibr 137	. 2 |- ( x e. A -> x e. ( A u. B ) )
4	3	ssriv 2949	1 |- A C_ ( A u. B )

Syntax hints:   \/ wo 629   e. wcel 1393   u. cun 2915   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931

This theorem is referenced by:  ssun2  3107  ssun3  3108  elun1  3110  inabs  3168  reuun1  3219  un00  3263  undifabs  3300  undifss  3303  snsspr1  3512  snsstp1  3514  snsstp2  3515  prsstp12  3517  sssucid  4152  unexb  4177  dmexg  4596  fvun1  5239  dftpos2  5876  tpostpos2  5880  ac6sfi  6352  ressxr  7069  nnssnn0  8184  un0addcl  8215  un0mulcl  8216  bdunexb  10040